A didactical fact is ‘any event that has a place and a time in the becoming of a mathematical instruction process and that, for some reason, is considered as a unit’ (Wilhelmi, Font, &Godino, 2005). We will say that it is a meaningful didactic fact (MDF) ‘if the didactic actions or practices that compose it play a role, or admit an interpretation, in terms of the intended instructional objective’ (Godino, Rivas, Arteaga, Lasa, & Wilhelmi, 2014) To support the description and understanding of the categories of affective entities, we will apply the notion of MDF to two experiences made in different educational and research contexts. On the one hand, we start from the descriptive and naturalistic research (Hernández, Fernández, & Baptista, 2010; Kelly, Lesh, & Baek, 2008) of an instructional process on probability implemented with a group of 18 students of third year of secondary education (14-15 years-old, from a public secondary education institute in Spain (Beltrán-Pellicer, & Godino, 2017)). The data collection instrument, from which the MDF are extracted, is the classroom journal of the teacher, the first author of this article and who acts as teacher-researcher. The second experience was also carried out by the first author in a public adult education centre. The students attend secondary education, or some of the courses of preparation for the acquisition of key competences or access to the degrees of vocational training, constituting a diverse sample of 38 people between 18 and 58 years-old. The collected data includes student’s productions in their notebooks and a free essay entitled ‘Mathematics and I, my relationship with mathematics so far’, an instrument used by other authors (Di Martino & Zan, 2011) to analyse attitudes and beliefs. The selected MDF from both data sources allow us to illustrate specific aspects of the affective domain, as we will see below, through the OSA categories.
Primary affective entities
Following the OSA pragmatic epistemological assumptions, we are now asking for the affective meaning of certain signs (in the sense of Peirce’s representamen), in any of the possible registers and representations, which may be verbal or written expressions, observable behaviours, etc. Such meaning must be sought in the systems of practices that a person performs to solve a problem situation, or towards a practice, an object, a mathematical process, or any mathematics study situation. There is agreement, within the scope of research in mathematical education, that the affective domain consists of three components: emotions, attitudes and beliefs. The origins of this classification go back to McLeod (1992) and, in this article, we will use this ontology of affective objects, to which we will add the values, construct included in the model of DeBellis & Goldin (2006).
Affective situations
It is necessary to consider a specific type of situation that provides the appropriate framework for describing affective practices. When a student is confronted with a situation-problem, an affective situation occurs that juxtaposes itself with the cognitive one, and which comes to include the purely personal meanings about it, in the form of emotions, attitudes, beliefs or values. For example, a mental block emotion towards a kind of problem-situation, a persevering attitude that facilitates the implementation of problem-solving heuristics, or a specific belief about the nature of the mathematical objects involved. In fact, all problem-situations in which the student’s active participation is required are strongly affective. Once the situation has been exposed, the personal beliefs of each student come into play, either to mathematics as a subject of study or to the context in which the proposed situation is framed. However, affective situations do not arise solely in response to a problem-situation, since the teaching and learning ecosystems provide constant reference points for the affective domain. In this way, there are situations of production, communication or, simply, of individual mathematical study. For example, a class session itself can bring up beliefs that influence the student’s attitude that day, without the need for any problem-situation yet. Therefore, it is feasible to describe an affective configuration for each of these situations, which will capture the circumstances of each component of the affective domain: emotions, attitudes, beliefs and values. Since we are interested in the relationships between affect and mathematical learning, we will confine affective situations to the circumstances in which ‘mathematical content’ is involved. The teacher may pose situations in which, specifically, the students’ beliefs are brought into play towards a concrete mathematical object. For example, the MDF1 reflects a situation being proposed by the teacher, in the first session of the lesson, to detect the students’ beliefs about chance and random sequences. MDF1: Situation-problem specifically implemented so that the students’ beliefs towards a specific mathematical object become evident. [Teacher’s diary] I introduce the first activity, about the distinction between random and deterministic phenomena. It consists of cutting 20 more or less equal pieces of paper, or 20 balls. Once students are finished, they are told to place them randomly on the table. Thus, in a dialogical environment and with manipulative material, the teacher observes the random dispositions of the objects on the tables and can ask questions to encourage reflection on them, while assessing the starting beliefs.
Affective practices
Affective practices are any action or affective manifestation that accompanies any mathematical practice. They can be manifestations about emotions, attitudes, beliefs or values about the objects put into play. Each of these affect expressions can vary in intensity throughout a practice or even disappear, giving rise to new manifestations. The great part of the affective trajectory remains hidden from the eyes of the teacher, because not all the affective states are manifested. Besides, it is not possible for a single person to observe the whole group to interpret small gestures or signs of every student. Nevertheless, an observation record, as a classroom diary (Porlán and Martín, 1991), helps to collect data on which to reflect later. And, in addition, there are instruments that can be incorporated into the teaching practice to gather information about the affective domain. This is the case, for example, of the humour map of the problems (Gómez-Chacón, 2000b), which the authors have used in previous research (Beltrán-Pellicer, 2015; Beltrán-Pellicer & Godino, 2017). Each student draws pictograms among 14 possible (or makes marks on a worksheet), to express what they feel during the process of solving a problem or task. The 14 pictograms represent 14 emotions: curiosity, great, boredom, indifference, mental block, despair, tranquillity, animation, haste, bewilderment, wracking my brain, pleasure, fun and trust. This map pursues a double purpose. On the one hand, it is a meta-affective practice, in which students become aware of their own emotional dynamics when trying to solve a mathematical situation. On the other hand, the information can be collected by the teacher, so that it can highlight affective facts that have allowed progress in the resolution and reflect on those that block or hamper progress. This tool was introduced in one of the experiences we refer to, as shown in MDF2:
MDF2: Introduction of the humour map. Considering the affective domain. Teacher’s diary […] let’s do the activities with the humour map of the problems. They are surprised when I give them the worksheets and proceed to explain them calmly. Likewise, I explain that they should mark those states with which they feel most identified.
Intervening and emerging objects
Although the categorisation of the affective domain in emotions, attitudes and beliefs is accepted by the research community, to which values can be added, the meaning of such constructs is still a matter of controversy. To describe and catalogue the affective objects that intervene or emerge in mathematical practices, we will use the tetrahedral model proposed by DeBellis and Goldin (2006), in which the meanings of the affective constructs are described as follows (p. 135): • Emotions: quickly changing feelings experienced in a conscious way or occurring pre-consciously or unconsciously during mathematical (or other) activity. Emotions vary from mild to intense and are locally and contextually immersed. • Attitudes: describe orientations or predispositions towards certain sets of emotional sensations (positive or negative), in particular (mathematical) contexts. This differs from the more common view of attitudes as predispositions toward certain patterns of behaviour. Attitudes are moderately stable, implying an interactive balance between affection and cognition. • Beliefs: they imply the attribution of some kind of truth or external validity to the system of propositions or other cognitive configurations. Beliefs are often highly stable, largely cognitive and structured, in which emotions and attitudes intersect with them, contributing to their stabilisation. • Values: including ethical and moral components, refers to personal truths or commitments deeply appreciated by individuals. They help to motivate long-term decisions or set short-term priorities. They can be highly structured, building value systems. Given the interaction with the cognitive domain, it may be convenient to consider, as a category of affective objects, the various modes of expression of the affects: gestures, terms of ordinary language, etc. (Álvarez, 2012), which would constitute the ostensible facet of affections. Emotions, attitudes, beliefs and values are relative to mathematical situations and practices, and to the distinct primary mathematical objects. It makes sense, therefore, to research the affective components towards the demonstrations, the procedures, the representations, etc. Figure 1 summarises the primary affective-cognitive categories.
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The characteristics of affective languages, which could be considered as a fifth category of affective objects, expand the semiotic registers and representations that emerge from the practices, since much of the affective charge is expressed nonverbally, within a system of information transmission, in which each element is interpreted by the different agents involved (teacher, students). Emotions, therefore, can arise as an instantaneous emotional response to a sensorial stimulus, which may have a mathematical character (a field of problems) or not (going to school). Although this distinction seems trivial, the origin of emotions is complex to interpret. Consider the MDF3, in which the annotation in the teacher’s diary indicates that his students show nervousness and agitation because of the proximity of a written test: MDF3: Nervousness because of the proximity of a written evaluation test. [Teacher’s diary] I tell the students that they already know the subject and that the examination date will be in 7 days. They complain, arguing that they have another test that day. Most of them, who were quite distracted and talking about other things, join in the discussion. Whereas it is clear that this is an emotional response, instantaneous, to a particular stimulus (the teacher announcing the examination date), the actual origin of the emotion could be based on specific beliefs about math tests or more general beliefs about school. Other times, it is easier to identify the source of those emotions, although it is not possible to establish absolute certainty, which would require more data collection tools. For example, the MDF4 describes an emotional reaction that could be named curiosity, to comments of the teacher that try to confront the students’ beliefs about random experiments with a specific mathematical fact, in this case, the stability of the relative frequencies. Some students hold this emotion, which leads to an attitude of interest: MDF4: Emotion that arouses interest. [Teacher’s diary]. I briefly introduce the stability of the relative frequencies, summarising the previous day activity (coin tosses). I ask them about the results of the coins, ‘to what number the percentage comes up’, to which they say that ‘one goes up and the other one goes down’. I see that at least A7 looks interested. I suggest them to think about what would happen if the coins were tossed 1000 times. The feedback system formed by the different components of the affective domain is put into play in any type of situation. The MDF5 shows how, when facing a problem-situation, some students show mental block emotion, while others start from a passive attitude, which they have been able to reach from the previous emotion or from causes unrelated to the situation. One of the teacher objectives in these cases is to intervene in the feedback loop (attitude-emotional) to promote progress in the teaching-learning process: MDF5: Emotion and attitude within a situation-problem. [Teacher’s diary] I see that there are students who are finishing, but also some who go slowly, either because they do not want to do it (case of A6) or because they get stuck. In the case of A6, I urge her to finish it. The students to whom the teaching-learning sequences are directed present beliefs about the mathematical objects that make up the trajectory of the instructional process. In the case of probability, its different meanings (Batanero, 2005) must be negotiated from personal belief systems, as seen in MDF6, where students, used to solve similar problem- situations with other procedures, are reluctant to use a new one: MDF6: Belief about the procedure to solve a situation-problem. [Teacher’s diary] I see that some of them have already reached an exercise in which they get stuck, and I introduce the tree diagrams, as a helpful way to solve it. But I notice that some students are still trying to do it without using the diagram and without success. At other times, in a dialogic interaction environment, it is the teacher who decides to inquire directly about the students’ beliefs. The MDF7 shows an example of this, in which the teacher asks about the distinction between random and deterministic phenomena, an issue that relates to the perception of chance: MDF7: Beliefs about random phenomena. [Teacher’s diary] They have no problem to specify the sample space of these experiments (balls extraction, pushpin that is thrown to the ground) or to distinguish if they are random or deterministic. I take the time to ask if predicting tomorrow weather will be random or deterministic. Ethical and moral values differ from beliefs in which, while the latter constitute judgments of subjective truth from the logical or empirical point of view, values refer to purely personal choices (that which is good, or desirable) (Goldin, 2002). However, belief systems and value systems are closely related, and at times, it is difficult and inoperative, to isolate them. The MDF8 exemplifies how the commitment to the study process (a personal choice that constitutes a value) influences the learning trajectory, directly reducing the effective teaching time: MDF8: Value about commitment to the study process. [Teacher’s diary] They take a long time to come from recess. I must raise my voice and show myself authoritative, so they can take out a notebook and a book. A5 takes even longer, speaks and laughs with his mates and has not brought the book. On the other hand, affective languages deserve special attention, and this is reflected in the key place reserved for them in Figure 1. Language, in its different registers, constitutes not only a communicative vehicle, but, being formed by signs which are constantly interpreted, is a tool of signification. In the case of the affective domain, non-verbal communication plays a fundamental role (Johnson, 1999; Knapp, Hall, & Horgan, 2013). In the same way that pupils’ productions, both written (also in their different registers) and verbal, provide indicators about the cognitive domain, the transmission of much of the affective information is done through facial expressions, gestures, postures, movements, etc. Harris and Rosenthal’s (2005) meta-study shows how students improve in certain facets when the teacher’s non-verbal language includes immediacy signs, such as gesturing when speaking, not sitting behind his desk, looking at students while talking, smiling, using a tone that is not monotonous, etc. Thus, students show interest in the course and the teacher, pay attention and have the perception that they have learned a lot in class (Rocca, 2004). Likewise, the results of his study also show correlations between the teacher’s non-verbal language and students’ cognitive performance, although this is something that is under study (Witt, Wheeless, & Allen, 2004). All these affective languages match interaction patterns that can be encompassed into one of the following three dimensions (Rompelman, 2002): opportunity to respond in a climate of trust, possibility of feedback, and consideration towards people (respect). Harris and Rosenthal (2005) also point out the difficulty of investigating empirically in the classroom environment, due to the apparatus required to capture all non-verbal information. Other authors agree in this regard. Mitchell (2013) points out that, given the positive relationship between the teacher’s non-verbal language and student attitudes, it is important that the teacher is not only enthusiastic about content, but should also show that enthusiasm to have a positive impact in the learning of the students. This influence of the teacher in the students’ beliefs towards mathematics and that, in the end, influence the other affective components, is evident in the essay excerpt shown here: Mathematics was never my strong point, rather, my Achilles heel. I came to hate them when I was in High School, despite this, little by little I can understand them thanks to my perseverance and dedication. Depending on the syllabus, I pay more attention when I find it interesting, although if I get bored I disconnect. Also depending on the teacher, which plays a fundamental role in making learning easier. In the excerpt, the student mentions the importance of certain attitudes (constancy, perseverance) and of the emotions that are awakened by some content (boredom), as well as the role that the teacher plays to encourage interest.
